This was the kind of stroll Roberto Mancini must wish for in Manchester City's title defence. After 52 minutes Carlos Tevez had made it 3–0 against a weak Leeds United side and thoughts began to turn to Chelsea's visit here on Sunday.

To win had been the only thing that mattered to Mancini and his band of under-performers. The previous time the City congregation witnessed a victory was in the previous round, on 26 January, at Stoke City.

From the team that lost at Southampton last time out Joe Hart, Samir Nasri, Gareth Barry, Edin Dzeko, Gaël Clichy and Joleon Lescott were omitted and only Barry was injured. Hart is yet to feature in the FA Cup this season and Clichy was rested, so Dzeko, Nasri and Lescott can consider themselves dropped and fingered for the blame for that dismal display at St Mary's.

Supine is not the first word to be associated with any Leeds or Neil Warnock side yet this was how they entered and exited this match. Their best chance did not arrive until the 71st minute, when Aidan White finally pinged over a cross aimed at the danger area though, as was the script, this came to nothing.

The yards of space Leeds allowed their hosts meant that after quarter of an hour City had come at them in a ceaseless blue wave that yielded a 2-0 lead and the impression that it was tie over and the 2011 winners were already in the sixth round.

"You come to a place like this and you have got to hope you start well, so it wasn't the perfect start," Warnock said.

Right at the start a flowing move involving Yaya Touré and James Milner outmanoeuvred Leeds down their left before Pablo Zabaleta crossed from the right into Jamie Ashdown's hands. Next, in the fifth minute, came Touré's opener, as this time City sliced through Leeds with a couple of one-twos. The Ivorian swapped passes with first David Silva, then Carlos Tevez, who with the outside of a boot played Touré in, and he made no mistake for his first City goal for two months.

Sergio Agüero, whose mother Adriano and son Ben were in an executive box, claimed the first of two goals from a penalty won by himself. After moving towards the right of Ashdown's goal, the Argentinian turned Tom Lees, who pulled him back. There appeared little in the challenge but Mark Clattenburg was in no doubt: he pointed to the spot and Agüero converted.

Warnock said: "It was never a penalty in a million years. I don't think it's a foul anywhere on the pitch. He stopped running and appealed and [the referee] fell for the three-card trick. I think Mark will be disappointed, the level that he is [as an official]. Tom was distraught."

The only Leeds player who was primed for a scrap was Michael Brown, who took turns to hack at Touré, Javi García, Tevez and Zabaleta. He may have lacked technical merit but he did take the fight to City.

This, then, made it a surprise when Brown was substituted at half-time, for David Norris though Warnock may have calculated that the midfielder would not have lasted the distance if he continued the foul-on-sight approach.

Tevez had Agüero to thank for his 10th goal of the season. After cutting in from the left his compatriot exchanged passes with Silva and drove into the Leeds area before flipping in a ball that Tevez forced home from close range.

Agüero's second of the afternoon – City's fourth – was a further illustration of how Leeds slumbered. Silva floated in a delivery to the forward, who was virtually unmarked in front of Ashdown's goal and he slotted in calmly for his 13th goal of the campaign.

Afterwards, Aguero tweeted of his son: "Did Ben ask me to score for him again...? Twice! I made good of it, esp. bc he's coming to spend his 4th bday with me. Love you, son!"

Jack Rodwell, a second-half replacement for García, also hit the crossbar with a header. But for Mancini this is a satisfactory win and one that will quieten his critics, until Sunday at least.

With Barnsley at home in the sixth round, the manager may yet sense a happy ending to this difficult season.